Waveform Optimization with Multiple Performance Metrics for Broadband Joint Communication and Radar Sensing

With most traditional hardware components replaced by digital processing modules, JCAS-enabled systems can reduce the number of connected devices and save the frequency resources [6, 7, 8]. To utilize the resources completely, most developed JCAS systems transmit single waveforms [9], except some works realizing JCAS functions via multiplexing technologies, such as time division multiplexing (TDM) [10], frequency division multiplexing (FDM) [11], or code division multiplexing (CDM) [12]. The resource utilization of single waveforms is highly efficient since the waveforms are shared by both communication and radar systems. Such waveforms can be developed from classical radar waveforms [13] or conventional communication waveforms [14].

Single waveform optimization has been reported in the literature. In [15], a multi-beam approach was proposed to flexibly generate JCAS sub-beams using analog antenna arrays. The optimization of the multi-beams was further investigated in [16]. This approach can adapt to varying channels but is suboptimal. In [17], the authors separated antenna arrays into two groups for realizing dual-function JCAS systems. The radar waveform falls into the nullspace of the downlink channel, such that the interference between radar signals and communication signals can be minimized. A multi-objective function was further applied to trade off the similarity between the generated waveform and the desired one in [18]. The multi-objective function in [18] is a weighted sum of two individual optimal waveforms. In [19], the authors maximized the radar signal-to-interference-plus-noise ratio (SINR) with a given specific capacity of communication channels. The work in [20, 21] further introduced a sub-sampling matrix for radar as an objective function of the optimization. It is noted that all these works studied waveform optimization in a narrowband scenario and ignored the frequency diversity.

When optimizing JCAS waveforms, different performance metrics for both communications and radar sensing can be used. The metrics for multiuser communication systems include multiuser interference (MUI) [18, 22] and effective channel gain (ECG) [23]. The MUI tolerance was analyzed for multi-input-multi-output orthogonal-frequency-division-multiplexing (MIMO-OFDM) JCAS systems in [22], using the interleaved signal model in [24]. The individual optimal communication waveform in [18] is also based on minimizing the MUI. However, simply minimizing MUI can cause the JCAS signals to fall into the nullspace of individual communication signals and result in a low SINR. To tackle this issue, the authors in [18] introduced an expected diagonal matrix that requires prior knowledge about the ECG.

As for radar sensing, typically considered performance metrics include mutual information (MI) [25, 26], Cramér-Rao bound (CRB) [9, 27], and minimum mean-squared error (MMSE) [28]. In [25], MI for an OFDM JCAS system was studied, and the power allocation on subcarriers was investigated by maximizing the weighted sum of the MI of radar and the MI of communication. In [26], the authors developed an MI measure that jointly optimizes the performance of radar and communication systems that overlap in the same frequency band. The CRB and MMSE are also commonly used metrics for waveform optimization. In [27], the authors derived the performance bounds in a single antenna system. In [28], the authors derived performance bounds for the radar estimation rate based on MMSE estimation bounds.

Although the above-mentioned metrics have been individually studied for JCAS systems, no links between them have been identified so far. Some of those papers can be invalid when the number of sensing targets is not equal to that of the communication users. For communications, the SINR is the main concern, while most papers do not address the SINR directly and use some indirect metrics. For radar sensing, we note that different metrics can be used but there are no parallel comparisons. Based on the limitations of prior works, we aim to build a common method that can utilize different metrics for both communications and radar sensing.

This paper aims to develop waveform optimization algorithms for broadband JCAS systems and establish connections between different metrics of optimization. We consider a practical JCAS system of deployment. JCAS systems, particularly those underlying communication signals, confront the major challenge of full-duplex, that is, transmitting and receiving at the same time using the same frequency channel [29]. Solutions bypassing full-duplex requirements were discussed in [30, 31]. Here, we adopt a single receive antenna dedicated to radar sensing, which is a low-cost and practical solution at the moment.

The key idea is to optimize a developed novel lower bound of SINR for communications under different constraints of the radar metrics. By studying optimization with respect to these metrics together, for the first time, we disclose the important connections and relationships between these metrics in JCAS systems.

The main contributions of this paper are summarized as follows.

• •

  We derive multiple metrics, including MUI and ECG for communications, and MMSE, MI, and CRB for radar sensing, in a broadband JCAS system. In the adopted system, bypassing the full-duplex requirement, the base station (BS) uses one single receive antenna for collecting the signals reflected from targets.

• •

  We develop a novel lower bound of SINR as the communication metric, which contains both MUI and ECG. The developed lower bound simplifies the expression of SINR and makes it flexible for joint waveform optimization, involving both communication and radar metrics.

• •

  For radar sensing, we derive multiple metrics, including MMSE, MI, and CRB. We show the connections between optimizing these metrics.

• •

  We apply these multiple performance metrics and formulate two kinds of JCAS waveform optimization problems by minimizing the proposed communication metric under the constraint of the radar metric. We obtain the corresponding closed-form solutions under certain conditions. When the conditions are not satisfied, we propose iterative solutions based on Newton’s method.

Notations: $\rm\bf a$ denotes a vector, $\rm\bf A$ denotes a matrix, italic English letters like $N$ and lower-case Greek letters $\alpha$ are a scalar. $|{\rm\bf A}|,{\rm\bf A}^{T},{\rm\bf A}^{H},{\rm\bf A}^{*},$ and ${\rm\bf A}^{\dagger}$ represent determinant value, transpose, conjugate transpose, conjugate, and pseudo inverse, respectively. We use ${\rm diag}({\bf a})$ to denote a diagonal matrix with diagonal entries being the entries of ${\bf a}$. $\|{\rm\bf A}\|_{F}$ and $[{\rm\bf A}]_{N}$ represent the Frobenius norm and the $N$th column of a matrix, respectively.

The sum rate or the SINR is the main concern for communications. However, the SINR has a non-convex form and can be difficult to optimize, especially when JCAS functions are required. In JCAS waveform optimization problems, both MUI and ECG have a dominant impact on the sum rate of communications. The MUI is on the denominator of SINR and can be expressed as

where ${\bf H}[k]=[{\bf h}^{\rm C}_{1}[k],\cdots,{\bf h}^{\rm C}_{U}[k]]$, ${\bf p}_{u}[k]$ is the $u$th column of ${\bf P}[k]$, and $I_{k,u}$ denotes as the MUI of the $u$th MU on the $k$th subcarrier. Traditional optimization methods that mitigate MUI for communications only are not appropriate for JCAS problems, since only minimizing MUI can cause that the precoding vectors fall into the nullspace of ${\bf H}[k]$, and result in a low ECG.

The ECG of MU-MISO systems is on the nominator of the SINR and can be written as

where $S_{k,u}$ denotes the ECG of the $u$th MU on the $g$th subcarrier. To make the following JCAS problems flexible and easy to solve, we develop a regulated bound of SINR as

where $\tilde{\bf H}_{u}[k]=\left[{\bf h}^{\rm C}_{1}[k],\cdots,{\bf h}^{\rm C}_{u-1}[k],{\bf h}^{\rm C}_{u+1}[k],\cdots,{\bf h}^{\rm C}_{U}[k]\right]$, ${\bf R}_{u}[k]=\mu\tilde{\bf H}_{u}[k]\tilde{\bf H}_{u}^{H}[k]-{{\bf h}^{\rm C}_{u}}[k]({{\bf h}^{\rm C}_{u}}[k])^{H}$, and $\mu$ is a weighting coefficient to balance between ECG and MUI.

The defined regulated bound, $J$, can be seen as a lower bound of SINR when two conditions are satisfied. The first condition is $S_{k,u}>\mu I_{k,u}$. The second condition is $I_{k,u}+U\sigma_{1}^{2}/P\leq 1$. Generally, $U\sigma_{1}^{2}/P$ can be neglected at high SNRs. Hence, both conditions can be satisfied at high SNRs by letting ${\bf P}[k]=({\bf H}[k])^{\dagger}$, that is, we need $I_{k,u}$ to be zero.

Theorem 1 unfolds that the defined metric, $J$, is a lower bound of SINR as long as the SNR is sufficiently large and the scaled MUI is less than the ECG. We also note that, when the noise term is zero, the SINR is no smaller than $KU\log_{2}(1+\mu)$. Hence, $\mu$ should be much larger than one at high SNRs to guarantee a high SINR.

By using $J$ to jointly optimize MUI and ECG, the individual waveform optimization problem for communications is formulated as

The minimization of $-J$ is still a non-convex problem since ${\bf R}_{u}[k]$ is not semi-definite, but we note that $J$ has a much simpler expression than the SINR. Hence, When multiple metrics including both communication and radar are considered, $J$ can be a good substitute for the SINR. By performing the eigen-decomposition of ${\bf R}_{u}[k]$, i.e., ${\bf R}_{u}[k]={\bf V}_{u}[k]{\bf E}_{u}[k]{\bf V}_{u}[k]^{H}$, we note that there exists one negative diagonal entry in ${\bf E}_{u}[k]$. The optimal solution of ${\bf p}_{u}[k]$ should be the eigenvector with the corresponding eigenvalue being negative.

MI and MMSE are widely-used metrics in radar systems. Maximizing MI enables the maximization of information in the received signal, which contains the parameters of targets during its propagation. Minimizing MMSE is to obtain the parameters with the highest accuracy. With obtaining the accurate information about the targets, the parameters related to sensing can be extracted using various sensing algorithms.

The MI of the received OFDM symbols for sensing can be represented as

From (10), it is clear that maximizing MI is a convex problem and the optimal ${\bf P}[k]$ satisfies that ${{\bf h}^{\rm S}}[k]=[{\bf U}_{\rm P}[k]]_{1}$, where ${\bf U}_{\rm P}[k]$ is the left singular matrix of ${\bf P}[k]$, that is, any ${\bf P}[k]$ satisfying $[{\bf U}_{\rm P}[k]]_{1}={{\bf h}^{\rm S}}[k]$ is the optimal solution. The globally optimal ${\bf P}[k]$ should maximize the first singular value of ${\bf P}[k]$. Hence, the optimal MI precoder has a rank of one. To make the precoder of dimension $N\times U$ have only one rank, the optimal MI precoder is obtained as

where ${\bf\Lambda}[k]$ is an arbitrary $1\times U$ matrix satisfying $\|{\bf\Lambda}[k]\|_{F}^{2}=\frac{U}{\|{{\bf h}^{\rm S}}[k]\|_{F}^{2}}$. Note that the size of the precoder is determined by the number of users instead of number of targets, since the number of targets can be varied from time to time.

The MMSE can be written as

where ${\hat{\bf h}}^{\rm S}[k]$ is the estimated radar channel. Referring to the derivations in [32], when ${{\bf h}^{\rm S}}[k]$ is Gaussian distributed, we have the result of $\frac{\partial{\rm MI}}{\partial P}=\frac{1}{2}{\rm MMSE}$. Hence, the larger the transmit power is, the less the MMSE becomes. Due to the existence of noise, the MMSE reaches its minimum value that is larger than zero. Since MMSE is related to different sensing algorithms, we will use MI instead of MMSE as one radar metric for the following JCAS problem formulation.

CRB is a theoretical lower bound of MMSE. Note that CRB is not up to the sensing algorithms and generally cannot be achieved. In our adopted broadband MISO systems, we mainly consider the accuracy of estimates for delays, complex gains, and AoDs of the targets. The Doppler frequency is assumed to be zero in the training period. Note that the BS system setup is not a typical mono-static radar. We assume that it is an approximate mono-static radar for the convenience of deriving the CRB. We rewrite the received radar signal of (5) into the form of vector, i.e.,

where ${\bf q}[k]={\bf P}^{T}[k]{\bf A}^{*}{\bf X}^{*}{\bf W}^{*}[k]{\bf 1}$ denotes the equivalent radar channel. Note that ${\bf q}[k]$ contains all parameters of interests, i.e., delays, complex gains, and AoDs, and contains the precoder to be determined. For convenience, we represent all parameters collectively using one common real-valued vector ${\bf\Theta}=\big{[}{\rm Re}[\alpha_{1},\cdots,{\alpha}_{L}],{\rm Im}[\alpha_{1},\cdots,\alpha_{L}],\tau_{1},\cdots,\tau_{L},{\Omega}_{1},\cdots,{\Omega}_{L}\big{]}^{T}$. It is noted that there are $4L$ parameters in total. Hence, ${\bf q}[k]$ is a function with $4L$ variables.

The CRB matrix, denoted as ${\bf B}$, contains all lower bounds of the estimates. The $(i,j)$th entry of ${\bf B}$ denotes the impact of the parameter $i$ on parameter $j$. The total CRB can be expressed as the Frobenius norm of ${\bf B}$. The CRB matrix is the inverse matrix of Fish information matrix (FIM), i.e., ${\bf B}={\bf F}^{-1}$. Referring to the derivations of [33] and using the so-called DOD/DOA Model defined in [33], we obtain our corresponding FIM as

where ${\bf F}_{1},\cdots,{\bf F}_{5}$, and ${\bf F}_{6}$ are given by Appendix B. The derivation can be referred to [33].

From Appendix B, we note that the CRB matrix depends on the covariance matrix, ${\bf Q}[k]={\bf P}[k]{\bf P}^{H}[k]$, only. Therefore, the CRB-based optimization problem is equivalent to optimizing ${\bf Q}[k]$. As we mentioned above, the total CRB equals the Frobenius norm of ${\bf B}$, which is equivalent to the sum of the singular values of ${\bf B}$. We aim to minimize the largest eigenvalue of the CRB matrix as in [34]. Minimizing the largest eigenvalue of the CRB matrix is equivalent to maximizing the smallest eigenvalue of the FIM. The waveform optimization under the eigen-optimization criterion can be directly formulated as a semi-definite-programming (SDP),

where $t$ is an auxiliary variable. However, the problem above is still difficult to address. We attempt to simplify the problem of (III-C). From Appendix B, we note that the norm of ${\bf F}_{4}$ has an order of $k^{2}$, which is significantly larger than that of other matrices in ${\bf F}$. Hence, we can approximately assume that ${\bf F}$ equals ${\rm Re}[{\bf F}_{4}]$, that is, only the accuracy of delays is counted.

Remark 1: Comparing MI and CRB, we note that they correspond to two types of optimization metrics. In the MI metric, the optimal waveform matrix can be obtained, while in the CRB metric, the optimal covariance waveform matrix can be obtained. For JCAS problem formulation, these two types of metrics also appear in many other metrics, such as channel capacity, desired semi-definite covariance matrix [18], and waveform similarity [35]. Hence, we will use these two types of optimal radar metrics and the proposed communication lower bound to formulate JCAS problems.

In this subsection, we constrain the MI of radar and simultaneously maximize the proposed lower bound of SINR, i.e., $J$. Since the optimal MI is obtained via section III-B, we can define a threshold of MI, $\tilde{H}_{0}$, that is smaller than the maximal MI. The MI-constrained JCAS optimization problem is formulated as

Note that minimizing the objective function above is not a convex problem, since ${\bf R}_{u}[k]$ in $J$ is not a semi-definite matrix. Hence, the problem above cannot be solved directly using convex-optimization toolboxes.

Since MI maximization is a convex problem, the range of ${\bf P}[k]$ that satisfies $\tilde{H}({{\bf h}^{\rm S}}[k];{\bf y}[k]|{\bf S}[k])\geq H_{0}$ should be a convex set. Hence, we can directly transform the MI constraint set into another convex set. For the convenience of following proposed algorithm, we adopt the convex set that limits the Euclidean distance between the JCAS precoder and the optimal MI precoder, i.e.,

where $\rho$ is a threshold to constrain the Euclidean distance between the JCAS precoder and the optimal MI precoder. The transformed constraint is a complex sphere.

For the third constraint, we note that it has the same expression as the term in the objective function. As long as we can find an initial value of ${\bf P}[k]$ satisfying $\mu I_{k,u}-S_{k,u}<0$, we can use iterative solutions to make the objective function keep dropping.

For the fourth constraint, we note that the norm of ${\bf H}[k]$ can be an arbitrary value. Hence, as long as $U\sigma^{2}/P<1$, we can make $I_{k,u}\leq 1-U\sigma_{1}^{2}/P$ by suppressing the norm of ${\bf H}[k]$. Hence, we omit the fourth constraint and transform the JCAS optimization problem of (IV-A) into

where ${\bf P}_{\rm MI}^{\star}[k]={{\bf h}^{\rm S}}[k]{\bf\Lambda}[k]$. Note that all variables are subcarrier dependent. For notational simplicity, we omit the parameter $k$ in the following of this subsection.

We define the two constraint functions in (IV-A) as $\psi({\bf P})=\|{\bf P}\|_{F}^{2}=\sum\limits_{u=1}^{U}\|{\bf p}_{u}\|_{F}^{2}$ and $\psi^{\prime}({\bf P})=\|{\bf P}-{\bf P}_{\rm rad,MI}^{\star}\|_{F}^{2}=\sum\limits_{u=1}^{U}\|{\bf p}_{u}-{{\bf h}^{\rm S}}\lambda_{u}\|_{F}^{2}$, where $\lambda_{u}$ is the $u$th diagonal entry of ${\bf\Lambda}$.

Before optimizing ${\bf P}$, we note that ${\bf\Lambda}$ in $\psi^{\prime}({\bf P})$ is still undetermined. It is clear to see that $\psi({\bf P})=U$ and $\psi^{\prime}({\bf P})=\rho$ are two complex spheres with the sphere center being ${\bf 0}$ and ${{\bf h}^{\rm S}}{\bf\Lambda}$, respectively. The objective function of $J$ can be seen as a saddle surface with the saddle point being the original point. The gradient of $J({\bf P})$ is given by

We determine ${\bf\Lambda}$, such that the absolute value of gradient is maximized, i.e.,

By using Least Square method, ${\bf\Lambda}$ is obtained as $\lambda_{u}^{\star}=c\|{\bf R}_{u}{{\bf h}^{\rm S}}\|_{F}$, where $c$ is a scaling factor, such that $\|{\bf\Lambda}^{\star}\|_{F}^{2}=\frac{U}{\|{{\bf h}^{\rm S}}\|_{F}^{2}}$.

The objective function in (IV-A) is a quadratic optimization problem with convex constrained sets. Such problems can be iteratively solved by using Newton’s method [36]. Besides, we also obtain the closed-form solution, as demonstrated in Theorem 2 and Corollary 1.

According to Theorem 2, we obtain an important corollary that can directly obtain the closed-form solution of ${\bf P}$, which is illustrated as follows.

However, the closed-form solution exists only when $f_{u}$ exists. In more general cases, we can obtain the following corollary.

Next, we propose an iterative algorithm based on Newton’s method [36], when the closed-form solution cannot be obtained. According to the proof of Theorem 2, we can make ${\bf p}_{u}$ reach one surface of the constraints. When ${\bf p}_{u}$ is on the surface of constraints, we aim to make the objective function be further decreased. Since there are two surfaces and only one will be reached, we consider two cases as follows.

In this subsection, we analyze the computational complexity of Algorithm 1. The main computation tasks in Algorithm 1 include the eigen-decomposition of ${\bf R}_{u}$ and the iterations. The complexity of the eigen-decomposition of ${\bf R}_{u}$ can be given by $\mathcal{O}(N^{3})$. Since there are $UK$ ${\bf R}_{u}[k]$, the total complexity of eigen-decomposition is $\mathcal{O}(N^{3}UK)$. The main steps in the iterations are Step 8 for Case 1 and Step 15 for Case 2, which both have a complexity of $\mathcal{O}(N)$, due to the calculation of $a_{u{\pm}}$ and ${b_{u\pm}}$. Hence, the complexity of the iterations is $\mathcal{O}(NUKN_{it})$, where $N_{\rm it}$ is the number of iterations. The overall complexity can be given by $\max\left(\mathcal{O}(N^{3}UK),\mathcal{O}(NUKN_{it})\right)$.